\newproblem{lay:1_6_5}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 1.6.5}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
  An economy has four sectors: Agriculture, Manufacturing, Services and Transportation. Agriculture sells 20\% of its output to
	Manufacturing, 30\% to Services, 30\% to Transportation, and retains the rest. Manufacturing sells 35\% of its output to Agriculture,
	35\% to Services, 20\% to Transportation, and retains the rest. Services sells 10\% of its output to Agriculture, 20\% to Manufacturing,
	20\% to Transportation, and retains the rest. Transportation sells 20\% of its output to Agriculture, 30\% to Manufacturing, 20\% to Services
	and retains the rest.
	\begin{enumerate}[a.]
		\item Construct the exchange table of this economy.
		\item Find a set of equilibrium prices for the economy if the value of Transportation is \$10,00 per unit. 
		\item The Services sector launches a successful ``eat farm fresh'' campaign, and increases its share of the output from the Agricultural sector to 40\%,
		      whereas the share of the Agricultural production going to Manufacturing falls to 10\%. Construct the exchange table for this new economy.
		\item Find a set of equilibrium prices for this new economy if the value of Transportation is still \$10,00 per unit. What effect has the ``eat farm fresh''
		      campaign had on the equilibrium prices for the sectors of this economy?
	\end{enumerate}
}{
   % Solution
	\begin{enumerate}[a.]
		\item The exchange matrix is given by\\
			\begin{center}
				$E=\begin{pmatrix}
					0.20 & 0.20 & 0.30 & 0.30 \\
					0.35 & 0.10 & 0.25 & 0.20 \\
					0.10 & 0.20 & 0.50 & 0.20 \\
					0.20 & 0.30 & 0.20 & 0.30 \\
				\end{pmatrix}$
			\end{center}
			First row implies that the output of Agriculture is sold 20\% to Agriculture, 20\% to Manufacturing, 30\% to Services and 30\% to Transportation.
		\item In equilibrium the expenses of any of the sectors are equal to its incomes. If we construct the vector of sector values $\mathbf{v}$, we may express this relationship as\\
			\begin{center}
				$\mathbf{v}=E\mathbf{v}\Rightarrow (I-E)\mathbf{v}=\mathbf{0}$
			\end{center}
			Expanding the different elements
			\begin{center}
				$\left(\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}-\begin{pmatrix}
					0.20 & 0.20 & 0.30 & 0.30 \\
					0.35 & 0.10 & 0.35 & 0.20 \\
					0.10 & 0.20 & 0.50 & 0.20 \\
					0.20 & 0.30 & 0.20 & 0.30 \\
				\end{pmatrix}\right)\begin{pmatrix} v_A \\ v_M \\ v_S \\ v_T\end{pmatrix}=\begin{pmatrix}0\\0\\0\\0\end{pmatrix}$\\
				$\left(\begin{array}{rrrr|r}
						0.80 & -0.20 & -0.30 & -0.30 & 0 \\
					 -0.35 &  0.90 & -0.35 & -0.20 & 0 \\
					 -0.10 & -0.20 &  0.50 & -0.20 & 0 \\
					 -0.20 & -0.30 & -0.20 &  0.70 & 0 \\
				\end{array}\right) \sim 
				\left(\begin{array}{rrrr|r}
						1 & 0 & 0 & -1 & 0 \\
					  0 & 1 & 0 & -1 & 0 \\
					  0 & 0 & 1 & -1 & 0 \\
					  0 & 0 & 0 &  0 & 0 \\
				\end{array}\right)$
			\end{center}
			The solution of this homogeneous system is
			\begin{center}
				$v_A=v_M=v_S=v_T \quad \forall v_T\in\mathbb{R}$
			\end{center}
			In particular, since the problem states that $v_T=10$, we have $v_A=v_M=v_S=v_T=10$.
		\item The new exchange table is
			\begin{center}
				$E=\begin{pmatrix}
					0.20 & 0.10 & 0.40 & 0.30 \\
					0.35 & 0.10 & 0.25 & 0.20 \\
					0.10 & 0.20 & 0.50 & 0.20 \\
					0.20 & 0.30 & 0.20 & 0.30 \\
				\end{pmatrix}$
			\end{center}
		\item The new augmented matrix is
			\begin{center}
				$\left(\begin{array}{rrrr|r}
						0.80 & -0.10 & -0.40 & -0.30 & 0 \\
					 -0.35 &  0.90 & -0.35 & -0.20 & 0 \\
					 -0.10 & -0.20 &  0.50 & -0.20 & 0 \\
					 -0.20 & -0.30 & -0.20 &  0.70 & 0 \\
				\end{array}\right) \sim 
				\left(\begin{array}{rrrr|r}
						1 & 0 & 0 & -1 & 0 \\
					  0 & 1 & 0 & -1 & 0 \\
					  0 & 0 & 1 & -1 & 0 \\
					  0 & 0 & 0 &  0 & 0 \\
				\end{array}\right)$
			\end{center}
			Again the solution is 
			\begin{center}
				$v_A=v_M=v_S=v_T \quad \forall v_T\in\mathbb{R}$
			\end{center}
			So, the campaign has had no effect on the different prices.
	\end{enumerate}
}
\useproblem{lay:1_6_5}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
